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Feb 28, 2018 at 19:05 comment added Alexandre Eremenko In the case of negative curvature, this is a much older result than Troyanov (it goes back to Picard). A very clear proof is in M. Heins, On a class of conformal metrics, Nagoya Math. J. 21 1962 1–60.
Jun 18, 2013 at 6:53 vote accept Analysis Now
Jun 18, 2013 at 5:15 history answered Jean-Marc Schlenker CC BY-SA 3.0